We will have a learning seminar on Choquet Theory during LP3 and LP4 in 2025, roughly every second week on Wednesday 10-12 in MVL14 (start at 10:00 and then about 60-90 minutes). The first four lectures will be based on the book by Phelps and will be accessible to Master students who took Functional Analysis. The last four lectures will cover more advanced topics in connection to C*-algebras.
Schedule and Topics
Lecture 1. Overview of Choquet theory
(Hannes Thiel 29.01.2025)
Sections 1+2 in [Phe01]
Lecture 2. Existence of representing measures
(Eusebio Gardella 19.02.2025)
Theorem of Choquet (existence in the metrizable case), Section 3 in [Phe01]
Theorem of Choquet-Bishop-de Leeuw (existence in general), Section 4 in [Phe01]
If times permits, also Theorems of Rainwater and Haydon, Section 5 in [Phe01]
Lecture 3. The Choquet boundary
(Anna Rohova 26.02.2025)
Section 6 in [Phe01] and selection of results in Sections 7,8,9 in [Phe01]
Lecture 4. Uniqueness of representing measures
(Martin Raum 12.03.2025)
Definition and characterizations of Choquet simplices, Section 10 in [Phe01]
If time permits, Section 11 in [Phe01]
Lecture 5. States on partially ordered groups
(Jan Gundelach, 09.04.2025 – date TBC)
Dimension groups and the Effros-Handelman-Shen theorem, Section 4 in [Goo86]
State space of partially ordered group, Section 6 in [Goo86]
Choquet simplices via interpolation groups, in particular Corollary 10.6 in [Goo86]
Characterization of Choquet simplex K in terms of Aff(K), in particular Theorem 11.4 in [Goo86]
Lecture 6. Traces on C*-algebras
(Guillaume Bellier, date TBA)
Tracial states on a unital C-algebra form a Choquet simplex.
Thoma (1964) # Über unitäre Darstellungen abzählbarer, diskreter Gruppen [Math. Ann. 153]
see also Theorem 11.4 in [BH82]
Every metrizable Choquet simplex arises from a simple AF-algebra.
Blackadar. Traces on simple AF C-algebras. J. Functional Analysis 38 (1980), 156-168. see also Moodie, Robert. Cones of traces arising from AF C-algebras. Doc. Math. 28 (2023), 1279-1321
Lecture 7. Dimension functions and quasitraces on C*-algebras
(Paolo Boldrini, date TBA)
Correspondence between dimension functions and quasitraces, see [BH82] and Section 6 in: Gardella, Perera. The modern theory of Cuntz semigroups of C-algebras. Preprint, arXiv. Blackadar-Handelman conjectures Theorem 4.1 in: Antoine, Perera, Robert, Thiel. C-algebras of stable rank one and their Cuntz semigroups. Duke Math. J. 171 (2022), 33-99.
Lecture 8. Noncommutative Choquet simplices
(Georg Huppertz, date TBA)
Overview on this new theory, based on:
Kennedy, Shamovich. Noncommutative Choquet simplices. Math. Ann. 382 (2022), 1591-1629.
Davidson, Kennedy. Noncommutative Choquet theory: A Survey. preprint, arXiv.
Literature
[Phe01] Phelps. Lectures on Choquet’s theorem. Lecture Notes in Math. 1757, Springer 2001.
[Alf71] Alfsen. Compact convex sets and boundary integrals. Ergeb. Math. Grenzgeb. 57, Springer 1971.
[Goo86] Goodearl. Partially ordered abelian groups with interpolation. Math. Surveys Monogr. 20, American Mathematical Society 1986.
[BH82] Blackadar, Handelman. Dimension functions and traces on C*-algebras. J. Functional Analysis 45 (1982), 297-340.